Constructions of generalized bent–negabent functions Aditi Kar Gangopadhyay and Ankita Chaturvedi Department of Mathematics, Indian Institute of Technology Roorkee Roorkee 247667 INDIA, {ganguli.aditi, ankitac17}@gmail.com

Abstract. In this paper, we consider generalized Boolean functions from Zn 2 to Zq . We describe two constructions of generalized bent–negabent functions.

Keywords: Generalized Boolean functions, nega–Hadamard transform, bent– negabent functions.

1

Introduction

Let Z, R and C denote the set of integers, real numbers and complex numbers respectively. Any element x ∈ Zn2 can be written as an n-tuple (xn , . . . , x1 ), where xi ∈ Z2 for all i = 1, . . . , n. The addition over Z, R and C is denoted by ‘+’. The addition over Zn2 for all n ≥ 1, is denoted by ⊕. Addition modulo q is denoted by ‘+’ and is understood from the context. If x = (xn , . . . , x1 ) and y = (yn , . . . , y1 ) are two elements of Zn2 , we define the scalar (or inner) product, respectively, the intersection by x · y = xn yn ⊕ · · · ⊕ x2 y2 ⊕ x1 y1 , x ∗ y = (xn yn , . . . , x2 y2 , x1 y1 ). The√ cardinality of the set S is denoted by |S|. If z = a + b ı ∈ C, then |z| = a2 + b2 denotes the absolute value of z, and z = a − b ı denotes the complex conjugate of z, where ı2 = −1, and a, b ∈ R. The conjugate of a bit b will also be denoted by ¯b. A function from Zn2 to Z2 is said to be a Boolean function on n variables and the set of all such functions is denoted by Bn . For more detail theory of Boolean functions we refer [1–3]. A function from Zn2 to Zq (q, a positive integer) is said to be a generalized Boolean function on n variables [10] and we denote the set of all such functions by Bnq . The (generalized) Walsh–Hadamard transform of f ∈ Bnq at any point u ∈ Zn2 is the complex valued function defined by X n Hf (u) = 2− 2 ζ f (x) (−1)u·x . x∈Zn 2

2πı

where ζ = e q is the q-th primitive root of unity. A function f ∈ Bnq is said to be generalized bent if and only if |Hf (u)| = 1 for all u ∈ Zn2 . The nega–Hadamard transform of f ∈ Bn at any vector u ∈ Zn2 is the complex valued function X n Nf (u) = 2− 2 (−1)f (x)⊕u·x ıwt(x) . x∈Zn 2

A function f ∈ Bn is said to be negabent if and only if |Nf (u)| = 1 for all u ∈ Zn2 . For further study of nega-Hadamard transforms and negabent functions we refer to [5–7, 11, 8, 13]. Definition 1. The nega–Hadamard transform of f ∈ Bnq at any point u ∈ Zn2 is defined by X n Nfq (u) = 2− 2 ζ f (x) (−1)u·x ıwt(x) . x∈Zn 2

Definition 2. A function f ∈ Bnq is a generalized negabent function if |Nfq (u)| = 1 for all u ∈ Zn2 . We shall use throughout the well-known identity (see [4]) wt(x ⊕ y) = wt(x) + wt(y) − 2wt(x ∗ y).

2

(1)

Existence of a subclass of generalized bent–negabent functions in generalized Maiorana–McFarland class

Schmidt [12] provided the generalization of Maiorana–McFarland construction of Z2 -valued bent functions to Z4 -valued bent function. The generalization is given in Lemma 1. St˘ anic˘ a et al. [9] generalized it for Zq -valued (for any even q) bent function. They refereed it by generalized Maiorana–McFarland class (GMMF). The GMMF is given in Lemma 2. Lemma 1. [12, Thm. 5.3] Let g : Zn2 → Z4 be arbitrary. Any function f : Z2n 2 → Z4 be given by f (x, y) = 2σ(x) · y + g(x), for all x, y ∈ Zn2 where σ is a permutation on Zn2 , then f is generalized bent Boolean function. Lemma 2. [9, Thm. 9] Suppose q is an even positive integer. Let σ be a permutation on Zn2 , let g : Zn2 → Zq be an arbitrary function, then the function f : Z2n 2 → Zq defined as q y · σ(x) for all x, y ∈ Zn2 (2) f (x, y) = g(x) + 2 is a generalized bent Boolean function.

Let σ is weight-sum invariant permutation on Zn2 , i.e., it satisfies by the following relation, wt(x ⊕ y) = wt(σ(x) ⊕ σ(y)) for all x, y ∈ Zn2

(3)

Noticed that if σ is orthogonal, i.e., σ(x) = A·x where A is orthogonal (i.e., AT = A−1 ), then it satisfies the imposed condition (since wt(σ(x) ⊕ σ(y)) = wt(A(x ⊕ y)), it suffices to show that wt(Az) = wt(z); for that, consider wt(Az) = (Az)T · (Az) = zT (AT A)z = wt(z)).
Theorem 1. Any function, f (x, y) = 2q y · σ(x) + g(x), for all x, y ∈ Zn2 , as in (2) on Z2n 2 is generalized bent–negabent if and only if g is generalized bent. Proof. We evaluate q

ζ g(x)+( 2 )y·σ(x) (−1)x·u⊕y·v ıwt(x)+wt(y)

X

Nfq (u, v) = 2−n

(x,y)∈Z2n 2

X

= 2−n

x∈Zn 2

X

= 2−n

X

ζ g(x) ıwt(x) (−1)x·u

(−1)(σ(x)⊕v)·y ıwt(y)

y∈Zn 2

ζ g(x) ıwt(x) (−1)x·u 2n/2 ω n ı−wt(σ(x)+v)

x∈Zn 2

= 2−n/2 ω n

X

ζ g(x) (−1)x·u ıwt(x)−wt(σ(x)+v) .

x∈Zn 2

Now, using the fact that σ is a weight-sum invariant permutation, and by (1), we obtain wt(σ(x) ⊕ v) = wt(x ⊕ σ −1 (v)), wt(x) − wt(σ(x) ⊕ v) = −wt(σ −1 (v)) + 2wt(x ∗ σ −1 (v)), and ı2wt(x∗σ

−1

(v))

= (−1)x·σ

−1

(v)

,

which implies that Nfq (u, v) = 2−n/2 ω n ı−wt(σ

−1

(v))

X

ζ g(x) (−1)(u⊕σ

−1

(v))·x

x∈Zn 2

= ω n ı−wt(σ

−1

(v))

Hg (u ⊕ σ −1 (v)).

Consequently, |Nfq (u, v)| = |Hg (u ⊕ σ −1 (v))|, which implies our claim. t u

3

Construction of generalized bent–negabent functions

Schmidt et al. [13] described construction of bent–negabent functions. In this section, we provide the characterization of generalized bent–negabent functions for arbitrary positive even integer q. Let us define V be mn dimensional vector space over Z2 , so that, V = Vn ⊕ Vn ⊕ . . . ⊕ Vn . | {z } m−times

Let the generalized Boolean function f : V ⊕ V → Zq expressed as follows f (x1 , . . . , xm , y1 , . . . , ym ) q σ(x1 , . . . , xm ) · (y1 , . . . , ym ) = g(x1 , . . . , xm ) + 2 where q is positive even integer, the function σ : V → V is of the form

(4)

σ(x1 , . . . , xm ) = (ψ1 (x1 ), φ1 (x1 ) ⊕ ψ2 (x2 ), . . . , φm−1 (xm−1 ) ⊕ ψm (xm )) and g : V → Zq is defined by g(x1 , . . . , xm ) = h1 (x1 ) + h2 (x2 ) + . . . + hm (xm ). Here, ψ1 , . . . , ψm , φ1 , φm−1 are permutations on Vn and h1 , . . . , hm : Vn → Zq be arbitrary generalized Boolean functions, i.e., f reads as q f (x1 , . . . , xm , y1 , . . . , ym ) = h1 (x1 ) + ψ1 (x1 ) · y1 2 m   q X (φj−1 (xj−1 ) ⊕ ψj (xj )) · yj + hj (xj ) + 2 j=2 Since σ is a permutation. Therefore f is Generalized Maiorana–McFarland (GMMF) type generalized bent function. t u Lemma 3. [13, Equation 3, 4] Let m be a positive integer satisfying m 6≡ 1(mod 3), and let k be an integer satisfying 0 < k < m and k ≡ 0(mod 3) or (m − k) ≡ 1(mod 3). Then P (z2k ) = 2kn ω c δψ(z2k )⊕a ,

for some c ∈ Z8 , a ∈ Vn

and  Q(z2k ) =

2(m−k)n ω d (−1)b·φ(z2k ) , if m ≡ 0 (mod 3), (m−k)n d b·φ(z2k ) −wt(ψ(z2k )) 2 ω (−1) ı if m ≡ 2 (mod 3).

for some d ∈ Z8 , b ∈ Vn , where φ, ψ are permutations on Vn and ω = 8-th primitive root of 1.

1+ı √ 2

is an

P (z2k ) = 2k−2 Y

X

j=1 zj ∈Vn

(−1)(zj+1 +uj )·zj ıwt(zj )

X z2k−1 ∈Vn

(−1)(ψ(z2k )+u2k−1 )·z2k−1 ıwt(z2k−1 )

and Q(z2k ) = 2m Y

X

(−1)(zj−1 +uj )·zj ıwt(zj )

j=2k+2 zj ∈Vn

X

(−1)(φ(z2k )+u2k+1 )·z2k+1 ıwt(z2k+1 ) .

z2k+1 ∈Vn

t u

Theorem 2. Let m be a positive integer satisfying m 6≡ 1(mod 3), and let k be an integer satisfying 0 < k < m and k ≡ 0(mod 3) or (m − k) ≡ 1(mod 3). Let f be defined as in (4), where σ(x1 , . . . , xm ) = (x1 , x1 + x2 , xk−1 + ψ(xk ), φ(xk ) + xk+1 , . . . , xm−1 + xm ) and g(x1 , . . . , xm ) = h(xk ). φ, ψ are permutations on Vn and h : Vn → Zq be arbitrary generalized Boolean function. Then f is generalized bent–negabent. Proof. Define by relabeling z2j := xj and z2j−1 := yj for 1 ≤ j ≤ m, so that we have f (x1 , . . . , xm , y1 , . . . , ym ) =   2k−2 2m X q X zj · zj+1 + z2k−1 · ψ(z2k ) + zj · zj−1 + z2k+1 · φ(z2k ) . h(z2k ) + 2 j=1 j=2k+2

2πı q

Let ζ = e

. Now, compute

Nfq (u1 , . . . , u2m ) = 2−mn

X

ζ f (z1 ,...,z2m ) (−1)(z1 ,...,z2m )·(u1 ,...,u2m ) ıwt(z1 ,...,z2m )

z1 ,...z2m ∈Vn −mn

=2

X

ζ h(z2k ) (−1)z2k ·u2k ıwt(z2k ) P (z2k )Q(z2k ).

z2k ∈Vn

Using Lemma 3 we have Nfq (u1 , . . . , u2m )  c+d P h(z2k )+ q2 (z2k ·u2k +b·φ(z2k )) wt(z2k ) ı δψ(z2k )⊕a) , if m ≡ 0 (mod 3), ω z2k ∈Vn ζ P = c+d h(z2k )+ q2 (z2k ·u2k +b·φ(z2k )) ω δψ(z2k )⊕a if m ≡ 2 (mod 3). z2k ∈Vn ζ ( q −1 −1 −1 −1 h(ψ (a))+ (ψ (a))·u +b·φ(ψ (a)) ) ıwt(ψ (a)) , if m ≡ 0 (mod 3), 2k 2( ω c+d ζ = q −1 −1 −1 c+d h(ψ (a))+ 2 ((ψ (a))·u2k +b·φ(ψ (a))) if m ≡ 2 (mod 3). ω ζ (5) Therefore, |Nfq (u1 , . . . , u2m )| = 1 for all (u1 , . . . , u2m ) ∈ V2m n . Hence we conclude our result. t u Example 1. Take m = 2 and k = 1 in Theorem 2. Then f becomes q f (x1 , x2 , y1 , y2 ) = h(x1 ) + {y1 · ψ(x1 ) + φ(x1 ) · y2 + y2 · x2 }. 2 Thus we can construct generalized bent–negabent functions in 4n variables of degree ranging from 2 to n.

4

Conclusion

In this paper, we provide a construction on generalized bent–negabent functions. Moreover, we identify a class of generalized bent–negabent functions.

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